Mathematical Sciences: Dynamics and Patterns in Reaction Diffusion Equations

反应扩散方程可以产生各种各样的类模式解。这些包括行波解，稳定解，细胞模式，振荡解，甚至以混沌方式表现的解。主要研究者将考虑的所有系统都是作为生物或化学系统的数学模型出现的。该项目将考虑燃烧模型，催化表面的化学反应，相变和可激膜的破裂。本研究项目的主要目标是研究这些模式可能通过哪些数学机制产生，分析每种模式的稳定性，并了解每种模式对潜在物理问题的物理意义。在这个项目中要解决的问题提出了一些基本的数学问题。特别令人感兴趣的是在多个空间维度中产生的模式。大部分分析将关注高维模式如何从基本的一维对象（如平面波解和径向对称解）中分叉。简单的例子表明，分岔模式的结构可能非常复杂。

Reaction diffusion equations may give rise to a variety of pattern like solutions. These include traveling wave solutions, stable steady solutions, cellular patterns, oscillatory solutions, and even solutions which behave in a chaotic manner. All of the systems which the principle investigator will consider arise as a mathematical model of a biological or chemical system. This project will consider models for combusion, chemical reactions on catalytic surfaces, phase transitions, and bursting in excitable membranes. The primary goals of this research project are to investigate through which mathematical mechanisms these patterns may arise, analyse the stability properties of each pattern, and understand what physical significance each pattern has for the underlying physical problem. The problems to be addressed in this project raise a number of fundamental mathematical issues. Of particular interest are patterns arising in more than one space dimension. Much of the analysis will be concerned with how higher dimensional patterns bifurcate from basically one dimensional objects such as planar wave solutions and radially symmetric solutions. Simple examples demonstrate that the structure of the bifurcating patterns may be incredibly complex.